Four rigid bodies, each with the same mass and radius, are spinning freely with the same angular momentum. Which object requires the maximum work to stop it ?

Options

A. A solid sphere spinning about a diameter
B. A hollow sphere spinning about a diameter
C. A solid disc spinning about an axis perpendicular to the plane of the disc and passing through the centre.
D. A hoop spinning about an axis along a diameter

Answer

To determine which of the four rigid bodies requires the maximum work to stop it, we need to analyze the relationship between work done, kinetic energy, and moment of inertia.

### Step-by-Step Solution:

1. **Understanding Work Done and Kinetic Energy**:
- The work done to stop a rotating body is equal to the change in its kinetic energy. When stopping the body, the final kinetic energy is zero, so the work done \( W \) can be expressed as:
\[ W = -K_0 \]
- Here, \( K_0 \) is the initial kinetic energy of the body.

2. **Expression for Rotational Kinetic Energy**:
- The rotational kinetic energy \( K \) of a rigid body is given by:
\[ K = \frac{1}{2} I \omega^2 \]
- Alternatively, it can also be expressed in terms of angular momentum \( L \):
\[ K = \frac{L^2}{2I} \]
- Since all bodies have the same angular momentum \( L \), we can compare their kinetic energies based on their moments of inertia \( I \).

3. **Identifying the Moment of Inertia for Each Body**:
- The moment of inertia \( I \) for the four bodies is as follows:
- Solid Sphere: \( I = \frac{2}{5} m R^2 \)
- Hollow Sphere: \( I = \frac{2}{3} m R^2 \)
- Solid Disc: \( I = \frac{1}{2} m R^2 \)
- Hoop: \( I = m R^2 \)

4. **Comparing Moments of Inertia**:
- To compare the moments of inertia, we can express them with a common denominator. The values become:
- Solid Sphere: \( \frac{12}{30} m R^2 \)
- Hollow Sphere: \( \frac{20}{30} m R^2 \)
- Solid Disc: \( \frac{15}{30} m R^2 \)
- Hoop: \( \frac{30}{30} m R^2 \)

5. **Determining Work Done**:
- Since the work done \( W \) is inversely proportional to the moment of inertia \( I \):
\[ W \propto \frac{1}{I} \]
- The body with the smallest moment of inertia will require the maximum work to stop.

6. **Identifying the Body with Minimum Moment of Inertia**:
- From the calculated values:
- Solid Sphere: \( \frac{12}{30} m R^2 \) (minimum)
- Hollow Sphere: \( \frac{20}{30} m R^2 \)
- Solid Disc: \( \frac{15}{30} m R^2 \)
- Hoop: \( \frac{30}{30} m R^2 \)
- The solid sphere has the smallest moment of inertia.

7. **Conclusion**:
- Therefore, the object that requires the maximum work to stop it is the **solid sphere**.